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nLab > Latest Changes: Delta-generated topological space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJan 27th 2012
    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 30th 2015

    I added several recent references.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 30th 2015

    I have cross-linked with directed homotopy theory

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 18th 2018

    I corrected a Unicode problem in Jiří Rosický’s name.

    Curiously, his name does not redirect to the article Jiří Rosický, even though it is hyperlinked.

    Instead, clicking on the hyperlinks attempts to create a new page with the same name (!).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2018

    Thanks for the alert. I have now copy-and-pasted the name from the entry title into the entry. The rendering looks the same, but now the link works, so probably there was some unicode ambiguity at play, or the like. (?) A similar issue still happens with hyphens here and there.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 18th 2018
    • (edited Jun 18th 2018)

    Re #5: I think the nLab does not normalize Unicode characters. As a result, the same letter ř, say, can be encoded in two different ways: as a single code point and as r followed by a combining character.

    I suggest that we add a Unicode normalization step during the CGI processing of submitted forms.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 18th 2020

    Delta-generated spaces are cartesian closed.

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2020

    I have added (here) statement of the proposition that Delta-generated spaces are the fixed points of the adjunction between topological spaces and diffeological spaces

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2020

    I have also turned the single previous sentence about categorical properties into the statement of two Propositions with pointers to the references where the proofs are to be found.

    diff, v9, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 6th 2020

    Would it add anything worthwhile to describe Delta-generated spaces as those diffeological spaces for which the unit is an isomorphism?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2020
    • (edited Jun 6th 2020)

    I have added a remark (here) that Δ\Delta-generated is the same as “Euclidean generated”:

    For each nn the topological simplex Δ n\Delta^n is a retract of the ambient Euclidean space/Cartesian space ℝ n\mathbb{R}^n (as a non-empty convex subset of a Euclidean space it is in fact an absolute retract). Hence the identity function on Δ n\Delta^n factors as

    id:Δ top n↪i nℝ n⟶p nΔ top n. id \;\colon\; \Delta^n_{top} \overset{i_n}{\hookrightarrow} \mathbb{R}^n \overset{p_n}{\longrightarrow} \Delta^n_{top} \,.

    It follows that every continuous function ff with domain the topological simplex extends as a continuous function to Euclidean space:

    Δ top m ⟶f X i n↓ ↗ ∃ ℝ n \array{ \Delta^m_{top} &\overset{f}{\longrightarrow}& X \\ \mathllap{{}^{i_n}}\big\downarrow & \nearrow _{\mathrlap{\exists}} \\ \mathbb{R}^n }

    Therefore the condition that a topological space XX be Δ\Delta-generated (Def. \ref{DeltaGeneratedSpace}) is equivalent to saying that its topology is final with respect to all continuous functions ℝ n→X\mathbb{R}^n \to X out of Euclidean/Cartesian spaces.

    We might thus equivalently speak of Euclidean-generated spaces.

    diff, v12, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2020

    I have expanded the statement about convenience (here). Currently it reads as follows:

    The category of Δ\Delta-generated spaces (Def. \ref{DeltaGeneratedSpace}) is a convenient category of topological spaces in that:

    diff, v15, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 6th 2020

    Do we have an example that shows that Δ-generated spaces are not locally cartesian closed?

    The quasitopos of diffeological spaces is locally cartesian closed, and Δ-generated spaces are reflective inside all diffeological spaces. Does this tell us anything about local cartesian closedness of Δ-generated spaces?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2020

    I have added a remark (here) on the terminology “D-topological space” (as discussed in another thread here)

    diff, v19, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 15th 2021

    Redirects: Δ-generated space, numerically generated space, etc.

    diff, v22, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2021

    added this pointer, for relation to k-spaces:

    • Philippe Gaucher, Section 2 of: Homotopical interpretation of globular complex by multipointed d-space, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 (arXiv:0710.3553)

    diff, v23, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2021

    where the model category structure is mentioned (here) I have added the remark that its Quillen equivalence to TopTop factors through the model structure on kTopk Top.

    diff, v24, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2021
    • (edited Sep 30th 2021)

    I have moved the Properties-section “As a convenient category of topological spaces” from the first to the last in the list of subsections (now here), so that all the statements about diffeological homotopy type can be referred to, and then I added the remark that things become ever more convenient by further embedding into diffeological spaces, then smooth sets, then smooth ∞\infty-groupoids, and in such a way that the canonical shape modality of the latter still sees the correct homotopy type of all topological spaces.

    (This addition is copied over from part of a similar edit that I just made at convenient category of topological spaces as announced here.)

    diff, v27, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2021

    I have enhanced the discussion of the cartesian closure (here):

    Made explicit the pleasing fact that we have two equivalent formulations of the internal hom, corresponding to the left and to the right side of the defining idempotent adjunction, namely

    Maps DTop(X,Y)≃Cdfflg(Maps Top(X,Y)), Maps_{DTop}(X,Y) \;\simeq\; Cdfflg \big( Maps_{Top} ( X ,\, Y ) \big) \,,

    (which follows already from Vogt 1971) but also

    Maps DTop(X,Y)≃Dtplg(Maps Dfflg(X,Y)). Maps_{DTop}(X,Y) \;\simeq\; Dtplg \big( Maps_{Dfflg} ( X ,\, Y ) \big) \,.

    Shimakawa & Haraguchi actually prove something stronger, part of which says that at least if the domain is a CW-complex, then the correct homotopy mapping space is also given by the internal hom formed in diffeological spaces (this might also want to go there, but now I have stated it here):

    X∈CWComplex↪DTopSp⇒∀A∈DTopSpMaps Dfflg(X,A)≃CdfflgMaps Top(X,A). X \,\in\, CWComplex \hookrightarrow DTopSp \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \underset{ A \,\in\, DTopSp }{\forall} Maps_{Dfflg} \big( X ,\, A \big) \;\; \simeq Cdfflg \, Maps_{Top} \left( X ,\, A \right) \,.

    This follows by combining two separate statements they make (as I have tried to make explicit in the entry now).

    diff, v29, current

  20. I have added a reference to my paper “Left properness of flows” which provides some information about Δ\Delta-generated spaces

    Philippe Gaucher

    diff, v32, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2021

    Thanks!

    I have added (here and here) link also to the TAC page of the article.

    diff, v33, current

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